Measured Data
Measured Data
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Measurements are central to the study of science and chemistry in particular. Every measurement provides three pieces of information: the magnitude of the measurement, the uncertainty of the measurement, and the units the measurement was made in.
For example, the mass of a 100 mL glass beaker is measured as 51.034 grams on an electronic balance with an uncertainty of 1 milligram or 0.001 grams. In the Wolfram Language each measurement is represented using a combination of and .
The magnitude and uncertainty of the measurement are the arguments to and the unit is the second argument to :
Quantity[Around[51.034,0.001],"Grams"]
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The first nonzero digit in the uncertainty indicates where the last significant digit in the magnitude is. Since the first nonzero digit in the uncertainty appears in the thousandths place, the digits 5 and 1 along with 0, 3, and 4 are all significant. The trailing zero is insignificant and present only for improved typesetting.
The number of significant digits can be determined automatically:
ResourceFunction["SignificantDigits"][Quantity[Around[51.034,0.001],"Grams"]]
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5
When measurements are used in computations, the uncertainty is automatically propagated by the Wolfram Language. This means that significant figures are automatic as well since the resulting uncertainty determines the number of significant digits.
Compute the mass of 25 mL of water in the 100 mL beaker using the same balance:
Quantity[Around[77.020,0.001],"Grams"]-Quantity[Around[51.034,0.001],"Grams"]
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The uncertainty has increased by the expected factor of for two measurements made on a balance with uncertainty δ:
2
Quantity[Around[,δ],"Grams"]-Quantity[Around[,δ],"Grams"]
m
final
m
initial
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Multiple measurements on the same instrument can be entered quickly using a .
Quantity[Around[{123.456,2.345,0.050},0.001],"Grams"]
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,,
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